| import OAI.Probability.FreeIsing.Estimates.ClippedArtanhWeight |
| import OAI.Probability.FreeIsing.Ising.CanonicalArtanh |
| import OAI.Probability.FreeIsing.Ising.ContinuousRootResponse |
| import OAI.Probability.FreeIsing.Stochastic.StoppedIntegralLocality |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal Topology |
|
|
| namespace Problem367.ClippedArtanhWeight |
| open ArtanhMartingaleFormula PosteriorCompensatorStopping PosteriorStoppingTimes |
|
|
| theorem weight_eq_stopped_before_exit {Ω : Type*} |
| {M A : ℝ≥0 → Ω → ℝ} {ω : Ω} |
| (hc : Continuous (fun t => M t ω)) (hzero : M 0 ω = 0) |
| (n : ℕ) (t : ℝ≥0) (ht : (t : WithTop ℝ≥0) ≤ jointExit M A n ω) : |
| weight (threshold n) (M t ω) = |
| firstWeight (stoppedProcess M (jointExit M A n) t ω) := by |
| have hb := abs_stopped_posterior_le (A := A) hc hzero n t |
| rw [stoppedProcess_eq_of_le ht] at hb ⊢ |
| exact weight_eq_of_abs_le hb |
|
|
| end Problem367.ClippedArtanhWeight |
|
|
| namespace Problem367.IsingObservationFactor |
| open IsingObservationInnovation ArtanhMartingaleFormula |
| open PosteriorStoppingTimes PosteriorCompensatorStopping |
|
|
| def canonicalClippedWeight {d : ℕ} (beta : ℝ) (n : ℕ) |
| (t : ℝ≥0) (x : Paths d) : ℝ := |
| ClippedArtanhWeight.weight (threshold n) |
| (continuousPosterior beta (ConnectedHull.root d) t x) |
|
|
| lemma canonicalClippedWeight_progressive {d : ℕ} (beta : ℝ) (n : ℕ) : |
| IsStronglyProgressive (filtration d) (canonicalClippedWeight beta n) := by |
| intro T |
| exact (ClippedArtanhWeight.continuous_weight (threshold_nonneg n) (threshold_lt_one n)).comp_stronglyMeasurable |
| (continuousPosterior_progressive beta (ConnectedHull.root d) T) |
|
|
| lemma canonicalClippedWeight_bound {d : ℕ} (beta : ℝ) (n : ℕ) |
| (t : ℝ≥0) (x : Paths d) : |
| |canonicalClippedWeight beta n t x| ≤ 1 / (1 - threshold n ^ 2) := |
| ClippedArtanhWeight.abs_weight_le (threshold_nonneg n) (threshold_lt_one n) _ |
|
|
| theorem canonicalClippedWeight_continuous_ae |
| {d : ℕ} (hd : 3 ≤ d) {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) (n : ℕ) : |
| ∀ᵐ x ∂observationLaw μ, Continuous (fun t => canonicalClippedWeight beta n t x) := by |
| filter_upwards [(continuousPosterior_modification hd hμ hb hcrit (ConnectedHull.root d)).1] |
| with x hx |
| exact (ClippedArtanhWeight.continuous_weight (threshold_nonneg n) (threshold_lt_one n)).comp hx |
|
|
| theorem canonicalClippedWeight_mul_covariance_before_exit |
| {d : ℕ} (hd : 3 ≤ d) {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) (n : ℕ) : |
| ∀ᵐ x ∂observationLaw μ, ∀ t : ℝ≥0, |
| (t : WithTop ℝ≥0) ≤ canonicalPosteriorExit beta (ConnectedHull.root d) n x → |
| ∀ w, canonicalClippedWeight beta n t x * continuousRootCovariance hb w t x = |
| continuousRootResponse hb w t x := by |
| filter_upwards [(continuousPosterior_modification hd hμ hb hcrit (ConnectedHull.root d)).1, |
| continuousPosterior_initial_zero hd hμ hb hcrit (ConnectedHull.root d)] with x hc hz |
| intro t ht w |
| have hm := abs_stopped_posterior_le |
| (A := canonicalCovarianceCompensator beta (ConnectedHull.root d)) hc hz n t |
| change (t : WithTop ℝ≥0) ≤ jointExit |
| (continuousPosterior beta (ConnectedHull.root d)) |
| (canonicalCovarianceCompensator beta (ConnectedHull.root d)) n x at ht |
| rw [stoppedProcess_eq_of_le ht] at hm |
| exact ClippedArtanhWeight.weight_mul_covariance_eq (threshold_nonneg n) (threshold_lt_one n) hm rfl |
|
|
| theorem canonicalClippedWeight_uniformSum_eq |
| {d : ℕ} (hd : 3 ≤ d) {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) (n : ℕ) (T : ℝ≥0) (k : ℕ) : |
| let M := continuousPosterior beta (ConnectedHull.root d) |
| let τ := canonicalPosteriorExit beta (ConnectedHull.root d) n |
| BrownianGrid.uniformSum (stoppedProcess M τ) (canonicalClippedWeight beta n) T k |
| =ᵐ[observationLaw μ] |
| BrownianGrid.uniformSum (stoppedProcess M τ) |
| (fun t x => firstWeight (stoppedProcess M τ t x)) T k := by |
| filter_upwards [(continuousPosterior_modification hd hμ hb hcrit (ConnectedHull.root d)).1, |
| continuousPosterior_initial_zero hd hμ hb hcrit (ConnectedHull.root d)] with x hc hz |
| apply StoppedIntegralLocality.uniformSum_stopped_congr |
| intro t ht |
| exact ClippedArtanhWeight.weight_eq_stopped_before_exit hc hz n t ht |
|
|
| end Problem367.IsingObservationFactor |
|
|
| end |
|
|
| end OAI |
|
|
